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# Aerodynamics – Philosophy of Method of Characteristics

1 - Question

For which of these flows is methods of characteristics limited to?

a) Inviscid flow

b) Viscous flow

c) One – dimensional flow

d) Quasi – two – dimensional flow

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Answer: aExplanation: For computing supersonic steady inviscid flows, there are two major numerical techniques that are used. The first being the methods of characteristics which is an older technique limited to inviscid flows. The other method is the finite – difference method which is applicable for both viscous and inviscid flows.

2 - Question

Which of these methods is employed to find solution using method of characteristics?

a) Taylor series

b) Fourier series

c) McLaurin series

d) Laurent series

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Answer: aExplanation: The flow field properties such as the pressure, density at the discrete points in a grid (xy coordinate plane) are found out by expanding these properties in using Taylor series. If we know the pressure at some point (i,j), then the pressure at the point (i + 1,j) point can be computed using this series as follows: ui + 1,j = ui,j + ∂P∂xi,j ∆x + ∂2P∂x2i,jΔx22+∂3P∂x3i,jΔx33 + ⋯.

3 - Question

The numerical solution results in certain amount of truncation error.

a) True

b) False

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Answer: aExplanation: In an ideal case, with infinite grid points, the numerical solution is exact. But in real life, the number of grid point for computing the flow field properties is finite and thus, the higher order terms are usually neglected causing ‘truncation error’.

4 - Question

What causes ‘round off’ error in numerical solution?

a) Finite grid points

b) Rounding number to a significant figure

c) Solving nonlinear equation

d) Incoherent boundary conditions

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Answer: bExplanation: Numerical solutions are obtained using the computers which often rounds off every number to a certain significant digit. Each of these rounding off results in small errors. If we decrease the truncation errors, the round – off error increases and these errors is a function of step size ∆x.

5 - Question

Which of these is the compatibility equation along C– characteristic line?

a) du + dpρa = 0

b) du – dpρa = 0

c) dpuρa = 0

d) duρa = 0

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Answer: bExplanation: C– characteristic line is the path along which the governing partial equation can be reduced to the ordinary differential equation. The compatibility equation along C– line is: du – dpρa = 0 The compatibility equation along C+ line is: du + dpρa = 0

6 - Question

What is the slope of the right running wave characteristic equation?

a) u + a

b) u – a

c) 2u

d) 2a

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Answer: aExplanation: The right running wave has a characteristic line as C+ which has a positive slope. It’s slope is given by dydx = u + a, where a is the local speed of sound which is equal to γRT−−−−√. Its compatibility equation is given by du + dpρa = 0.

7 - Question

The Riemann invariants are constant along the characteristic line.

a) True

b) False

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Answer: aExplanation: Riemann invariants are obtained by integrating the two compatibility equations obtained along the characteristic lines C+ and C–. These are as follows: J+ = u + ∫dpρa = const J– = u – ∫dpρa = const J+ and J– are hence constant along the characteristic lines.

8 - Question

Which grid is used to find the flow field properties using finite difference solutions?

a) Rectangular grid

b) Non – rectangular grid

c) Square grid

d) Circular grid

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Answer: aExplanation: For computing the flow field properties at points in the flow, there is often grid considered in a plane with known properties at one of the grid point. Using that, properties at other grid points can be computed. In the method of characteristics, non – rectangular grid point is considered and for finite – difference solution, rectangular grid is considered.

9 - Question

What is the cause of truncation error in numerical solution?

a) Neglecting higher order terms

b) Rounding number to significant figure

c) Solving nonlinear equation

d) Incoherent boundary conditions

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Answer: aExplanation: In numerical solution, while expanding flow field properties in terms of Taylor series, higher order terms are often neglected. When these terms are neglected, it leads to errors known as truncation error.

10 - Question

Method of characteristics is applicable for which of these flows?

a) Inviscid, subsonic flow

b) Steady, supersonic flow

c) Unsteady, subsonic flow

d) Unsteady supersonic flow

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Answer: bExplanation: Method of characteristics is a type of numerical technique used today for computing flow field of steady supersonic flows. This method was developed in the year of 1920s which is a very classical numerical approach.

11 - Question

Where is the application of method of characteristics?

a) Optimizing the wing

b) Designing the compressor

c) Designing contour of nozzle

d) Optimizing the shape of the fuselage

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Answer: cExplanation: Despite the method of characteristics being highly intensive in terms of the computational load, the invention of high speed computers have made the possibility to use method of characteristics in designing the contour of the supersonic nozzle.